Question

A bucket that has a mass of 20 kg when filled with sand needs to be lifted to the top of a 15 meter tall building. You have a rope that has a mass of 0.2 kg/m. Once the bucket reaches the top of the building it only has mass 18 kg because there was a hole in the bottom and sand was leaking out at a constant rate while it was being lifted to the top of the building. Find the work done lifting the bucket (sand and rope) to the top of the building.

Answer 1

work done lifting the bucket (sand and rope) to the top of the building,

W=67.46 Nm

Step-by-step explanation:

in this question we have given

mass of bucket=20kg

mass of rope=
`.2(kg)/(m)`

height of building= 15 meter

We have to find the work done lifting the bucket (sand and rope) to the building =work done in lifting the rope + work done in lifting the sand

work done in lifting the rope is given as,
`W_(1)=Force * displacement`

=
`\int\limits^(15)_0 {.2x} \, dx` ..............(1)

=
`.1* 15^2`

=22.5 Nm

work done in lifting the sand is given as,
`W_(2)=Force * displacement`

`W_(2)=\int\limits^(15)_0 F \, dx`.................(2)

Here,

F=mx+c

here,

c=20-18

c=2

m=
`(20-18)/(15-0)`

m=.133

Therefore,

`F=.133x+2`

Put value of F in equation 2

`W_(2)=\int\limits^(15)_0 (.133x+2) \, dx`

`W_(2)=.133 * 112.5+2*15\\W_(2)=14.96+30\\W_(2)=44.96 Nm`

Therefore,

work done lifting the bucket (sand and rope) to the top of the building,
`W=W_(1)+W_(2)`

W=22.5 Nm+44.96 Nm

W=67.46 Nm